# Entropy and Entropy Production: Old Misconceptions and New Breakthroughs

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## Abstract

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## 1. Thermodynamic Entropy and Disorder. Is there a Difference in the Evolution of Animate and Inanimate?

_{V}and then calculating it based on the formula:

_{X}is the temperature in the state X in which the absolute value of the entropy S

_{X}is determined. In order to find the absolute value of entropy, e.g., according to Equation (2), another postulate, the third law of thermodynamics (at T ° 0, the entropy tends to zero), needs to be introduced.

_{1}= 1. Let us assume that the particles can move (diffuse) freely in the vessel. Obviously, the system will reach the second macroscopic state. This state can be implemented in three equiprobable ways (see Figure 1), and therefore Ω

_{2}= 3. According to Equation (3), the second state will show higher entropy as compared to the first state. This agrees with the second law: an isolated system changes from some prepared (nonequilibrium) state to the equilibrium state and the entropy increases. Obviously, the second state (Figure 1) is more disordered than the first one as the particles can be located in one part of the vessel or the other at different moments of time.

^{*}) and (4

^{**}) can have a considerably higher value, and then the total change of entropy dS/dt will be negative. As a consequence, the entropy of the body at hand will decrease. Let us note that the total entropy of the whole isolated system (the body with the surrounding environment) will not go down in complete agreement with the second law of thermodynamics. If entropy is viewed as a measure of disorder, then the body in the above example will mostly give the contained disorder to the environment rather than receive it from the outside or produce it internally. As a result of this exchange, the order inside the body increases. It is important to note that, according to Schrödinger, only the animate can have this property, and the inanimate has no such ability. The proposed mechanism of the entropy reduction and the Equation (4) provoke no objections. However, the division between the animate and inanimate systems based on their ability to “absorb order” from the environment (if it was really true) does not eliminate but, on the contrary, creates a barrier between the animate and the inanimate. Indeed, it is stated that the animate systems have some exclusive properties (qualitative differences), the special physics. Let us quote [4]: “New laws to be expected in the organism.”

_{2,}one of the most common minerals on Earth) and lower (for instance, as compared to 3.9 kJ/K or 6.8 kJ/K, the entropy of water and air, respectively).

## 2. Entropy Production. Direction of the Evolution of Inanimate and Animate Systems

_{i}(particularly, the heat flux) and the thermodynamic forces X

_{i}(particularly, the temperature gradient):

_{i}and X

_{i}as follows:

#### 2.1. Prigogine’s and Ziegler’s Principle

_{i}, be maintained constant, then the minimum entropy production density shall be the necessary and sufficient condition for the stationary state of the nonequilibrium system.

_{i}are preset, then the true thermodynamic fluxes J

_{i}satisfying the side condition (7) give the maximum value of the entropy production density σ(J). Mathematically, this principle can be written using the Lagrange multiplier µ in the form:

_{0}. Then, based on Ziegler's principle, the system will adjust its thermodynamic fluxes in order to maximize the entropy production. If the entropy production is a quadratic function, then such an adjustment will result in the linear relationship between the fluxes/forces, and the system will reach its stationary nonequilibrium state. If the behavior of this system is considered for longer (than τ

_{0}) times τ, then a number of thermodynamic forces can become free (varying). In this case, the fluxes formed according to Ziegler (Equations (5) and (6)) will start decreasing the thermodynamic forces, which, in turn, will reduce the fluxes and so on. As a result, the minimum entropy production is reached. Thus, some hierarchy of processes can be seen: for small times, the system maximizes the entropy production with the fixed forces and, as a result, the linear relations between the fluxes and forces become valid; for a large scale of time, the system varies the free thermodynamic forces in order to reduce the entropy production [23].

**Figure 2.**Geometrical interpretation of the principles of Ziegler

**(a)**and Prigogine

**(b)**for the case of two thermodynamic fluxes (forces) in the system. In the first graph, the maximum entropy production on the plane positioned angle-wise to the ordinate axis is sought for; and in the second graph, the minimum entropy production on the plane parallel to the ordinate axis is sought for.

#### 2.2. Maximum Entropy Production Principle (MEPP)

- (1)
- There is a well-known variation method of solving the linearized Boltzmann equation used in the study of the transfer in gases, metals, and semiconductors. It is less known that M. Kohler [26] and J. Ziman [27] showed the identity of this variation method with the following statement: the velocity distribution function for nonequilibrium gas systems is such that the entropy production density is a maximum at preset gradients of the temperature, the concentration and the mean velocity. There are generalizations of this statement that are valid both for the quantum systems and for the relatively dense systems [28,29].
- (2)
- Based on the analysis of a large amount of empirical data, M. Berthelot [30] formulated the following statement: if several chemical reactions can take place in a system without exposure to an external energy, the reaction, which is accompanied by the release of the largest heat, is realized. It can be shown [24] that for a very wide range of chemical processes (especially at relatively low temperatures) the amount of heat produced by the reaction per unit of time is directly proportional to the entropy production. Therefore, the observations of Berthelot can be viewed as a kind of experimental confirmation of the maximum entropy production principle.
- (3)
- (4)
- By analyzing the experimental and theoretical data of the nonequilibrium crystallization, D. Temkin [32], J. Kirkaldy [33], E. Ben-Jacob [34], etc. assumed that, in the case of the fixed supersaturation/supercooling, the growing dendrite will select the maximum possible rate. As is known (see, for example, [35,36]), the entropy production density of the system under consideration is proportional to the squared growth rate of the crystal. Therefore, the presented data is indicative of the maximum entropy production principle.

#### 2.3. The Range of MEPP Applicability

#### 2.4. МЕРР and Metastability

#### 2.5. On the MEPP Falsifiability, the Jaynes Maximization, and the MEPP Justification

#### 2.6. MEPP and the Biological Evolution

- (1)
- (2)
- (3)

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Martyushev, L.M.
Entropy and Entropy Production: Old Misconceptions and New Breakthroughs. *Entropy* **2013**, *15*, 1152-1170.
https://doi.org/10.3390/e15041152

**AMA Style**

Martyushev LM.
Entropy and Entropy Production: Old Misconceptions and New Breakthroughs. *Entropy*. 2013; 15(4):1152-1170.
https://doi.org/10.3390/e15041152

**Chicago/Turabian Style**

Martyushev, Leonid M.
2013. "Entropy and Entropy Production: Old Misconceptions and New Breakthroughs" *Entropy* 15, no. 4: 1152-1170.
https://doi.org/10.3390/e15041152